#149186

order	512
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#392
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11322
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,13,7,11,27,36,8,12,28,37,5,25,26,52)(3,14,18,40,19,31,46,57,17,39,45,61,16,9,29,32)(4,10,23,34,24,33,51,58,22,35,50,59,21,48,49,63)(15,41,43,62,44,30,54,56,20,38,47,60,42,55,53,64);(1,3,5,16,8,17,7,19)(2,9,11,14,12,31,25,39)(4,20,21,44,22,15,24,42)(6,18,26,29,28,45,27,46)(10,30,33,38,35,55,48,41)(13,32,36,40,37,57,52,61)(23,47,49,54,50,43,51,53)(34,56,58,60,59,64,63,62);(1,4)(2,10)(3,15,17,20)(5,21)(6,23)(7,24)(8,22)(9,30,31,55)(11,33)(12,35)(13,34)(14,38,39,41)(16,42,19,44)(18,43,45,47)(25,48)(26,49)(27,51)(28,50)(29,53,46,54)(32,56,57,64)(36,58)(37,59)(40,60,61,62)(52,63)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56382);